Hubble parameter data from Cosmic Chronometers. Has time dilation been taken into account?

The expansion history of the Universe can found by measuring the differential age evolution of cosmic chronometers. This yields a measurement of the Hubble parameter H(z) as a function of redshift.

The basic idea of the Cosmic Chronometer method is that the Hubble parameter is related to the differential redshift-time relation as $$H(z) = \frac{-1}{1+z}\times\frac{dz}{dt}\tag1$$

(1) is derived from

$$H(z)= \frac{\dot a}{a}$$

$$a=\frac{1}{1+z}$$ and

$$\frac{da}{dt}=\frac{da}{dz} \times \frac{dz}{dt}$$

Then the quantities $$dz$$ and $$dt$$ need to be measured for a passively evolving system such as a group of stars, $$dz$$ is found from spectroscopic measurements.

For $$dt$$ the ‘cosmic chronometer’ was a direct spectroscopic observable (the 4000 ˚A break) known to be linearly related with the age of the stellar population

and a plot of $$H(z)$$ against $$z$$ can be obtained like the one above.

“The solid line and the dashed regions...show the fiducial flat LCDM cosmology... H0 = 67.8 km/s/Mpc, m = 0.308. For comparison an Einstein-de Sitter model is shown...”

• Moresco et al DOI:10.1088/1475-7516/2016/05/014

The method doesn’t need absolute ages of stars, but the difference in ages of the stars.

The question is this:

Since both the actual age of a star and the difference in ages of a pair of stars are both subject to time dilation, has this properly been accounted for in the method?

For example, with easy numbers.

If time dilation caused a 10.2 million year old star at a redshift of $$z$$ to appear to be 5.1 million years old, and a 10 million year old star further away at redshift $$z+dz$$ to appear to be 5 million years old, the $$dt$$ would be measured as 0.1 million years, but in reality it should be 0.2 million years.

It doesn’t seem to be taken into account, or is it already there in equation (1)?

The method in summary is to observe two galaxies that were assumed to have formed at the same cosmic epoch but are observed at different redshifts. The difference in their redshift is $$\Delta z$$ and then by measuring some spectral feature that depends on age (in their own frame of reference), to estimate the difference in age between them $$\Delta t$$. The idea is to choose a pair of galaxies that formed at a considerably higher redshift than where they are now observed; where "considerably higher" I think means that the time since they formed is a lot longer than the observed age difference between them.
• @JohnHunter The light left the first star when it was 5.9 Gyr old (in its own frame of reference - that is what this value of time means). Therefore when that light gets to us, it looks like it was emitted by a star that was 5.9 Gyr old, but redshifted by a factor of $1+z$. There isn't any other correction to apply. Mar 28 '21 at 14:50